Generalized Overcubic Partitions

• Generalized overcubic partitions are a variant of integer partitions where even parts appear in multiple colors and overlined on first occurrence.
• They extend classical partition theory by integrating modular forms, q-series, and explicit generating functions to capture intricate congruence phenomena.
• Analytic techniques such as the circle method and Rademacher formulas yield precise asymptotics and reveal properties like log-concavity and density results.

A generalized overcubic partition of a nonnegative integer $n$ with color parameter $c \ge 1$ is a variant of classical integer partitions, where each even part may occur in $c$ distinct “colors” and each part (regardless of color) may be overlined upon its first occurrence. This combinatorial framework simultaneously extends the notions of overpartitions and colored (or “cubic”) partitions. The arithmetic of generalized overcubic partitions unifies and generalizes a spectrum of classical partition congruences, admits modular and $q$-series interpretations, and exhibits intricate congruence and density phenomena.

1. Definitions and Generating Functions

Let $\bar a_c(n)$ denote the number of generalized overcubic partitions of $n$ with color parameter $c \ge 1$. Formally, such a partition is one in which:

• Odd parts may appear at most once and may be overlined or not.
• Each even part of size $2j$ may appear in up to $c$ distinct colors, each color class permitting an overlining on its first occurrence.

The generating function for $\bar a_c(n)$ is

$$\sum_{n \ge 0} \bar a_c(n) q^n = f_1^2 f_2^{2c-3} f_4^{c-1},$$

where $f_r = \prod_{m \ge 1}(1 - q^{mr})$ (Das et al., 25 Mar 2025, Ghoshal et al., 4 Dec 2025). For $c=1$ this specializes to

$$\sum_{n\ge0} \bar a_1(n) q^n = \frac{(-q;q)_\infty}{(q;q)_\infty},$$

the generating function for ordinary overpartitions, while $c=2$ yields the classical cubic overpartition function.

For extensions such as overcubic partition pairs and triples, the generating functions take the form

$$\sum_{n \ge 0} \overline b_k(n) q^n = \frac{f_6^k}{f_1}$$

with $k=1$ corresponding to single overcubic partitions, $k=2$ to pairs, $k=3$ to triples, and so on (Saikia et al., 19 Oct 2024).

2. Rademacher-Type Formulas and Exact Asymptotics

Advanced analytic techniques (notably the circle method and Rademacher’s exact formula) have been adapted to compute $\bar a(n)$ for cubic overpartitions: $$\bar a(n) = \frac{3\pi}{16n\sqrt{2}} \sum_{k \ge 1,\ k\ \mathrm{odd}} \frac{A_k^{(1)}(n)}{k} I_2\Bigl(\frac{\pi}{k}\sqrt{\frac{3n}{2}}\Bigr) + \frac{\pi}{4n\sqrt{2}} \sum_{\substack{k \ge 1 \ k \equiv 2 \pmod{4}}} \frac{A_k^{(2)}(n)}{k} I_2\Bigl(\frac{\pi}{k}\sqrt{2n}\Bigr),$$ where $I_2$ is the Bessel function and $A_k^{(1)}, A_k^{(2)}$ are Kloosterman-type sums involving Dedekind sums (Agarwal et al., 27 Sep 2025). This series converges absolutely, supporting the extraction of precise asymptotics, error bounds, and limit properties such as log-concavity for large $n$.

3. Congruences and Density Results

3.1. Finite Modulus Congruences

For all $n \ge 1$, and general $c\ge1$,

$$\bar a_c(n) \equiv \begin{cases} 2 \pmod{4} & \text{if } n = k^2, \ 2(c+1) \pmod{4} & \text{if } n = 2k^2, \ 0 \pmod{4} & \text{otherwise}. \end{cases}$$

(Das et al., 25 Mar 2025, Ghoshal et al., 4 Dec 2025)

There exist further fine-grained congruences modulo 8 and 12, for progressions in $n$ and for $c$ in specified arithmetic classes. For example, for all $i\ge1$ and $n\ge0$: $$\bar a_{2i}(4n+3) \equiv 0 \pmod{4},\quad \bar a_{9i}(9n+6) \equiv 0 \pmod{12}$$ with various higher-power $2$-adic congruences for $c$ in progressions (Das et al., 25 Mar 2025, Paksok et al., 24 Mar 2025).

3.2. Infinite Families and Lacunarity

For any fixed $k\ge1$,

$$\lim_{X\to\infty} \frac{\#\{n\le X:\bar a_{c}(n)\equiv0\pmod{2^k}\}}{X}=1,$$

i.e., the sequence is highly lacunary modulo $2^k$: almost all $\bar a_{c}(n)$ are divisible by any fixed $2^k$. Similar density phenomena occur for certain odd primes $p$ and progressions in $c$ (Das et al., 25 Mar 2025, Ray et al., 2018).

Moreover, for $\lambda\ge1$, $m\ge0$, $t\ge1$, $n>0$: $$\overline a_{2^\lambda m + t}(n)\equiv\overline a_t(n)\pmod{2^{\lambda+1}},$$ yielding a hierarchy of exact congruences for large ranges of the color parameter $c$ (Paksok et al., 24 Mar 2025).

4. Combinatorial Proofs and Structural Results

Combinatorial arguments provide refined insight into congruences. The crucial observation is that with $r$ distinct parts in a partition, there are $2^r$ possible overlining choices; for $r\ge2$, this forces the overall count divisible by 4. For single-part partitions, calculation reduces to summing over the odd and even divisors of $n$, with factors of $2$ (odd) and $2c$ (even) reflecting color and overline possibilities. The combinatorial mod 4 congruence

$$\bar a_c(n)\equiv2\tau_{\mathrm{odd}}(n)+2c\tau_{\mathrm{even}}(n)\pmod{4}$$

where $\tau_{\mathrm{odd}}$ and $\tau_{\mathrm{even}}$ count odd and even divisors, confirms the analytic results structurally (Ghoshal et al., 4 Dec 2025).

5. Turán Inequalities, Log-Concavity, and Subadditivity

Analytic bounds derived from Rademacher-type formulas enable precise estimates for

$R(n) = \frac{\bar a(n+1)\bar a(n-1)}{\bar a(n)^2}$

showing for $n\gg1$ that $R(n)<1$ and thus strict log-concavity for almost all $n$, i.e.,

$$\bar a(n)^2 > \bar a(n-1)\bar a(n+1),\quad (n\ge10).$$

More generally, higher-order Turán (hyperbolicity) inequalities for the Jensen polynomials built from $(\bar a(n))$ are established by asymptotic analysis, confirming hyperbolicity for sufficiently large $n$ (Agarwal et al., 27 Sep 2025).

Log-subadditivity and generalized log-concavity,

$\bar a(n)\bar a(m)\geq \bar a(n+m)$

and

$\bar a(n)^2 > \bar a(n-m)\bar a(n+m)$

hold except for finitely many small exceptions, generalizing classical results due to Bessenrodt-Ono and DeSalvo-Pak for $p(n)$ (Agarwal et al., 27 Sep 2025).

6. Modular Forms, Dissections, and Methods

The generating functions for generalized overcubic partitions, especially for special $c$, can be written as integer-weight eta-quotients, making them excellent candidates for the modular forms machinery. Isolated and infinite family congruences are proved by:

• Constructing suitable eta-quotients and checking modular weights, levels, and characters,
• Applying Hecke operators and Sturm's theorem to assert global congruences from finitely many checks,
• Employing Radu’s algorithm for algorithmic verifications in higher prime-power moduli,
• Using classical $q$-series identities (e.g., 2- and 3-dissections of theta functions).

Combinatorial methods, such as divisor parity analysis and bijective overline colorings, provide alternative routes to certain congruences, especially mod $4$ and $8$, without recourse to modular forms (Ghoshal et al., 4 Dec 2025, Das et al., 25 Mar 2025, Amdeberhan et al., 15 Jun 2024, Paksok et al., 24 Mar 2025).

7. Generalizations, Extensions, and Open Problems

Numerous generalizations are active research fronts:

• $r$-colored overcubic partitions, with generating function

$$\sum_{n\ge0} \bar a_{\ell,r}(n)q^n = (−q;q)_\infty(−q^\ell;q^\ell)_\infty^{r−1} / \bigl[(q;q)_\infty(q^\ell;q^\ell)_\infty^{r−1}\bigr],$$

define weakly holomorphic modular forms on $\Gamma_0(\ell)$, leading to more intricate modular and arithmetic phenomena (Agarwal et al., 27 Sep 2025).

• Overcubic partition $k$-tuples are encoded by

$\sum_{n\ge0} b_k(n)q^n = \frac{f_6^k}{f_1}$

with congruences and lacunarity (almost all coefficients divisible by 2) holding for fixed $k$ (Saikia et al., 19 Oct 2024).

• Significant open questions remain about systematic elementary proofs of $p^2$-congruences, congruences for odd primes beyond $2,3,12$, and joint lacunarity across multiple moduli (Das et al., 25 Mar 2025, Saikia et al., 19 Oct 2024).
• There is an expectation (conjectural in some cases) of Rademacher-type formulas, log-concavity, Turán hyperbolicity, and subadditivity in the entire $(\ell,r)$-colored overcubic setting, with verification for finite $N(d)$ exceptions remaining open (Agarwal et al., 27 Sep 2025).

These structures establish generalized overcubic partitions as a central object in the arithmetic of partition theory, connecting combinatorics, modular forms, and explicit congruence phenomena.